2,293 research outputs found
Markov chains and optimality of the Hamiltonian cycle
We consider the Hamiltonian cycle problem (HCP) embedded in a controlled Markov decision process. In this setting, HCP reduces to an optimization problem on a set of Markov chains corresponding to a given graph. We prove that Hamiltonian cycles are minimizers for the trace of the fundamental matrix on a set of all stochastic transition matrices. In case of doubly stochastic matrices with symmetric linear perturbation, we show that Hamiltonian cycles minimize a diagonal element of a fundamental matrix for all admissible values of the perturbation parameter. In contrast to the previous work on this topic, our arguments are primarily based on probabilistic rather than algebraic methods
On transition matrices of Markov chains corresponding to Hamiltonian cycles
International audienceIn this paper, we present some algebraic properties of a particular class of probability transition matrices, namely, Hamiltonian transition matrices. Each matrix P in this class corresponds to a Hamiltonian cycle in a given graph G on n nodes and to an irreducible, periodic, Markov chain. We show that a number of important matrices traditionally associated with Markov chains, namely, the stationary, fundamental, deviation and the hitting time matrix all have elegant expansions in the first n−1 powers of P , whose coefficients can be explicitly derived. We also consider the resolvent-like matrices associated with any given Hamiltonian cycle and its reverse cycle and prove an identity about the product of these matrices
Topological localization in out-of-equilibrium dissipative systems
In this paper we report that notions of topological protection can be applied
to stationary configurations that are driven far from equilibrium by active,
dissipative processes. We show this for physically two disparate cases :
stochastic networks governed by microscopic single particle dynamics as well as
collections of driven, interacting particles described by coarse-grained
hydrodynamic theory. In both cases, the presence of dissipative couplings to
the environment that break time reversal symmetry are crucial to ensuring
topologically protection. These examples constitute proof of principle that
notions of topological protection, established in the context of electronic and
mechanical systems, do indeed extend generically to processes that operate out
of equilibrium. Such topologically robust boundary modes have implications for
both biological and synthetic systems.Comment: 11 pages, 4 figures (SI: 8 pages 3 figures
Hamiltonian cycles and subsets of discounted occupational measures
We study a certain polytope arising from embedding the Hamiltonian cycle
problem in a discounted Markov decision process. The Hamiltonian cycle problem
can be reduced to finding particular extreme points of a certain polytope
associated with the input graph. This polytope is a subset of the space of
discounted occupational measures. We characterize the feasible bases of the
polytope for a general input graph , and determine the expected numbers of
different types of feasible bases when the underlying graph is random. We
utilize these results to demonstrate that augmenting certain additional
constraints to reduce the polyhedral domain can eliminate a large number of
feasible bases that do not correspond to Hamiltonian cycles. Finally, we
develop a random walk algorithm on the feasible bases of the reduced polytope
and present some numerical results. We conclude with a conjecture on the
feasible bases of the reduced polytope.Comment: revised based on referees comment
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